prod1[NN_,k_,i_]:=Product[k*i-c,{c,0,NN-1}]

d[NN_,k_,l_]:=Sum[(((-1))^i)*prod1[NN,k,i]*l!/(i!*(l-i)!),{i,1,l}]

c[NN_,l_,j_,nu_,s_,k_]:=(((((-1))^(NN+j))*(E^(s/k))*(nu^l)*d[NN,k,l])/(j!(NN-1-j)!*l!*(s^(l-j/k))*(k^(1+j/k))))

moment[NN_,nu_,s_,k_]:=If[NN>0,Sum[c[NN,l,j,nu,s,k]*Gamma[l-j/k,s/k],{l,1,NN},{j,0,NN-1}],1]

NN=5; nu=2; s=10; k=1/5;

result=moment[NN,nu,s,k];

FullSimplify[result]

N[result,10]